Modeling dynamic systems, including fluid dynamic systems, using computers, particularly high-speed digital computers, is a well known and cost efficient way of predicting system performance for both steady thermophysical and transient conditions without having to physically construct and test an actual system. A benefit to computer modeling is that the effect on performance of changes in system structure and composition can be easily assessed, thereby leading to optimization of the system design prior to construction of a commercial prototype.
Known modeling programs generally use a “multi-cell” approach, where the structure to be modeled is divided into a plurality of discrete volume units (cells). Typically, the computer is used to compute thermophysical values of the fraction of the system within each cell, such as mass, momentum, and energy values, as well as associated fluid system design parameters such as density, pressure, velocity, and temperature, by solving the conservation equations governing the transport of thermophysical value units to or from the neighboring cells and/or through cell boundaries. For a fixed geometric system model using Cartesian coordinates, and absent a system boundary, each cell would have six cell neighbors positioned adjacent the six faces of the cube-shaped cell. An example of a computational fluid dynamics (“CFD”) modeling program is the MoSES Program available from Convergent Thinking, LLC, Madison, Wis. However, improvements are possible and desirable in existing modeling programs.
For conventional CFD programs the user initially must supply a three-dimensional grid representing the object to be simulated. For these CFD programs, the grid or “mesh” generation is the most user-intensive portion of the modeling process, especially for complex geometries with moving boundaries, where the grid may have to be reformed after each transient time step.
There are two types of boundary fitted grids. The more conventional type of boundary fitted grid morphs the cells near the boundary to conform to the shape of the geometry, e.g., a six-sided cell near a boundary would not necessarily be a perfect cube. The other method is commonly called a “cut-cell” method. In traditional “cut cell” methods boundary cells are cut one or more times to provide a better “fit” with solid surfaces of the fluid system being modeled. The solid surfaces can be represented by an array of triangles fitted to solid surface which are then associated with the individual model cells adjacent the solid surface. In traditional cut-cell methods there are two approaches to handling cells that are cut by multiple surface triangles. In one approach, the surface effects of each triangle are treated separately for each cut-cell. This process can result in longer CFD simulation run times. In the second method, the cuts of multiple triangles are approximated by a single planar cut. In this approach, the original geometry is not accurately represented because information has been lost in the surface approximation.
Cut-cell Cartesian methods can result in two types of problematic cells: “sliver” cells and “split” cells. Sliver cells are boundary cells that have a very small volume compared to the non-boundary cells. Cells with small volumes can require unreasonably small time steps or an unreasonable number of iterations, and therefore more computation, to keep the solution stable.
Split cells also are a problem for logical block structured grids, of which Cartesian is a type. In a Cartesian grid, each cell is shaped as a cube and is assumed to have only one neighbor in the direction of each of its six faces. Each cell has a logical (x, y, z) coordinate and each of its neighbors will differ by exactly one logical step. Much of the simplicity of Cartesian grids is a direct result of this structured connection of cells to their six neighbors. For this system to work properly there must be only one cell at a logical coordinate. If a surface of the fluid system to be modeled cuts a Cartesian cell into two or more parts, this presents a problem in conventional CFD codes because there will be two separated cells at the same location. These two separate cell parts have the same logical coordinates, thus making it impossible for their respective neighbors to locate the appropriate cell part by looking at the logical coordinates alone.
It should be noted that, in some cases, even with user intervention it is not possible to overcome these problems with the more traditional cut-cell approaches. The disclosed method and apparatus are directed to mitigating one or more of the problems set forth above.